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11: 22.4 Periods, Poles, and Zeros
Again, one member of each congruent set of zeros appears in the second row; all others are generated by translations of the form 2 m K + 2 n i K , where m , n . …
12: 34.2 Definition: 3 j Symbol
Either all of them are nonnegative integers, or one is a nonnegative integer and the other two are half-odd positive integers. They must form the sides of a triangle (possibly degenerate). … For alternative expressions for the 3 j symbol, written either as a finite sum or as other terminating generalized hypergeometric series F 2 3 of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).
13: Notices
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  • Index of Selected Software Within the DLMF Chapters

    Within each of the DLMF chapters themselves we will provide a list of research software for the functions discussed in that chapter. The purpose of these listings is to provide references to the research literature on the engineering of software for special functions. To qualify for listing, the development of the software must have been the subject of a research paper published in the peer-reviewed literature. If such software is available online for free download we will provide a link to the software.

    In general, we will not index other software within DLMF chapters unless the software is unique in some way, such as being the only known software for computing a particular function.

  • 14: 31.12 Confluent Forms of Heun’s Equation
    This has regular singularities at z = 0 and 1 , and an irregular singularity of rank 1 at z = . …
    15: 14.5 Special Values
    §14.5(v) μ = 0 , ν = ± 1 2
    14.5.22 Q 1 2 ( cos θ ) = K ( cos ( 1 2 θ ) ) - 2 E ( cos ( 1 2 θ ) ) ,
    14.5.23 Q - 1 2 ( cos θ ) = K ( cos ( 1 2 θ ) ) .
    16: 4.23 Inverse Trigonometric Functions
    Arctan z and Arccot z have branch points at z = ± i ; the other four functions have branch points at z = ± 1 . …
    §4.23(iv) Logarithmic Forms
    Other Inverse Functions
    Care needs to be taken on the cuts, for example, if 0 < x < then 1 / ( x + i 0 ) = ( 1 / x ) - i 0 . …
    17: 18.7 Interrelations and Limit Relations
    §18.7 Interrelations and Limit Relations
    Chebyshev, Ultraspherical, and Jacobi
    Legendre, Ultraspherical, and Jacobi
    §18.7(iii) Limit Relations
    18: 1.9 Calculus of a Complex Variable
    or in polar form ((1.9.3)) u and v satisfy … One of these domains is bounded and is called the interior domain of C ; the other is unbounded and is called the exterior domain of C . … or its limiting form, and is invariant under bilinear transformations. Other names for the bilinear transformation are fractional linear transformation, homographic transformation, and Möbius transformation. …
    19: About MathML
    In the future, we also hope to be able to deliver the Content form of MathML, a form which more faithfully preserves the semantic information of the mathematics. This should allow easier reuse of the mathematics, such as copying the formula into other documents, computer algebra systems or other computation engines. … A few other browsers have partial support, which is encouraging, but insufficient for DLMF’s complex material. …
    20: 31.11 Expansions in Series of Hypergeometric Functions
    For other expansions see §31.16(ii).
    §31.11(ii) General Form
    The expansion (31.11.1) for a Heun function that is associated with any branch of (31.11.2)—other than a multiple of the right-hand side of (31.11.12)—is convergent inside the ellipse . …